Abstract
Chapter 3 continues the discussion of ranks and cranks from Chapter 2. In his lost notebook, sometimes in an oblique fashion, Ramanujan provided the 2-, 3-, 5-, 7-, and 11-dissections for the generating function of cranks. These dissections can be expressed in either of two equivalent formulations – identities or congruences –, although the equivalence is not obvious. In his lost notebook, Ramanujan states one of the dissections as an identity, two as congruences, and two in anomalous ways, because only the quotients of theta functions appearing in the dissections are given. In this chapter, we consider the dissections as congruences and provide two different methods for obtaining proofs. One of them stems from an identity that is recorded cryptically by Ramanujan in his lost notebook and which is proved in Chapter 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106.
B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, Cranks and dissections in Ramanujan’s lost notebook, J. Combin. Thy., Ser. A 109 (2005), 91–120.
A.B. Ekin, Inequalities for the crank, J. Combin. Thy., Ser. A 83 (1998), 283–289.
A.B. Ekin, Some properties of partitions in terms of crank, Trans. Amer. Math. Soc. 352 (2000), 2145–2156.
R.J. Evans, Generalized Lambert series, in Analytic Number Theory (Allerton Park, IL, 1995), Vol. 1, B.C. Berndt, H.G. Diamond, and A.J. Hildebrand, eds., Birkhäuser, Boston, 1996, pp. 357–370.
F.G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77.
F.G. Garvan, The crank of partitions mod 8,9 and 10, Trans. Amer. Math. Soc. 322 (1990), 79–94.
M.D. Hirschhorn, A birthday present for Ramanujan, Amer. Math. Monthly 97 (1990), 398–400.
M.D. Hirschhorn, Winquist and the Atkin–Swinnerton-Dyer partition congruences for modulus 11, Australasian J. Comb. 22 (2000), 101–104.
V.G. Kač and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125–264.
V.G. Kač and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, in Lie Theory and Geometry, Birkhäuser, Boston, 1994, pp. 415–456.
W.-C. Liaw, Contributions to Ramanujan’s Theories of Modular Equations, Ramanujan-Type Series for 1/π, and Partitions, Ph.D. Thesis, University of Illinois at Urbana–Champaign, Urbana, 1999.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
J. Tannery and J. Molk, Éléments de la Théorie des Fonctions Elliptiques, Tome III, Gauthier-Villars, Paris, 1898.
L. Winquist, An elementary proof of p(11n+6)≡0(mod11), J. Comb. Thy. 6 (1969), 56–59.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Andrews, G.E., Berndt, B.C. (2012). Ranks and Cranks, Part II. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_3
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3810-6_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3809-0
Online ISBN: 978-1-4614-3810-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)